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CaKFe₄As₄ is an iron arsenide superconductor in which partial substitution of Fe by a transition metal shifts the ground state from superconducting to antiferromagnetically (AFM) ordered. Employing single-crystal neutron diffraction data we show that the AFM order is a non-collinear, commensurate structure with a hedgehog spin-vortex crystal (SVC) arrangement within the Fe planes and a simple AFM stacking perpendicular to them. The long-range SVC order coexists with superconductivity, however, the interplay of both phenomena depends on the kind of transition-metal doping: For electron-doping by Ni, a gradual suppression of the ordered magnetic moment is observed with decreasing temperature below the superconducting transition temperature. For hole-doping by Mn, the ordered magnetic moment increases smoothly with decreasing temperature and superconductivity seem not to influence the magnetic order.